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A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices

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Trends in Harmonic Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 3))

Abstract

We use bounded cohomology to define a notion of volume of an \(\operatorname{SO}(n,1)\)-valued representation of a lattice \(\varGamma<\operatorname{SO}(n,1)\) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff (Geom. Dedicata 117, 111–124 (2006)) in this setting. Our approach gives in particular a proof of Thurston’s version of Gromov’s proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov–Thurston proof using the simplicial volume invariant.

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Acknowledgements

Michelle Bucher was supported by Swiss National Science Foundation project PP00P2-128309/1, Alessandra Iozzi was partially supported by Swiss National Science Foundation project 2000021-127016/2. The first and third named authors thank the Institute Mittag-Leffler in Djurholm, Sweden, for their warm hospitality during the preparation of this paper. Finally, we thank Beatrice Pozzetti for a thorough reading of the manuscript and numerous insightful remarks.

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Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211, Geneva 4, Switzerland

    Michelle Bucher

  2. Department Mathematik, ETH, Rämistrasse 101, 8092, Zurich, Switzerland

    Marc Burger & Alessandra Iozzi

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  1. Michelle Bucher
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  2. Marc Burger
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Correspondence to Alessandra Iozzi .

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  1. Dipto. Matematica, Università di Roma - Tor Vergata, Via della Ricerca Scientifica 1, Roma, 00133, Italy

    Massimo A. Picardello

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Bucher, M., Burger, M., Iozzi, A. (2013). A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices. In: Picardello, M. (eds) Trends in Harmonic Analysis. Springer INdAM Series, vol 3. Springer, Milano. https://doi.org/10.1007/978-88-470-2853-1_4

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